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Is it possible for a category to have an initial object $i$ such that
$$\text{Hom}(a,i) \neq \emptyset, \forall a \in \text{Obj}$$
And also such that this initial object is not a final object? I know that the empty set is an initial object but not a terminal object in $\text{Set}$, but that's mainly due to the fact that there are no arrows terminating at the empty set at all right?
Yes, this is possible. For instance, consider the opposite category of nonempty sets. A singleton set $i$ is initial in this category (since it is terminal in the usual category of sets), but there is a map from every object to $i$ (since there is a function from $i$ to any nonempty set). And $i$ is not terminal, since if $a$ is any set with more than one element then there is more than one map $a\to i$.
However, if you assume that your category has a terminal object $t$ and an initial object $i$ with a map from every object, then $i$ must be terminal. For if $f:t\to i$ is any map, let $g:i\to t$ be the unique map (using the universal property of either $i$ or $t$). Then $fg$ and $gf$ must both be the identity, because there is only one map $i\to i$ and only one map $t\to t$. So $f$ and $g$ are inverse isomorphisms, so $i$ is terminal since $t$ is.
